Theorem ttcwf2 36695

Description: If a transitive closure class is a set, then it is well-founded, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcwf2	⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem ttcwf2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)))
2	1	eldifad 3897	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ TC+ 𝐴)
3		ttctr2 36664	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ TC+ 𝐴 → 𝑥 ⊆ TC+ 𝐴)
4	2, 3	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ TC+ 𝐴)
5		dfss2 3903	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ TC+ 𝐴 ↔ (𝑥 ∩ TC+ 𝐴) = 𝑥)
6	4, 5	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) = 𝑥)
7		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
8		inssdif0 4304	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On) ↔ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
9	7, 8	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On))
10	6, 9	eqsstrrd 3952	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ ∪ (𝑅1 “ On))
11		vex 3431	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
12	11	r1elss 9719	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On))
13	10, 12	sylibr 234	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ ∪ (𝑅1 “ On))
14	1	eldifbd 3898	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → ¬ 𝑥 ∈ ∪ (𝑅1 “ On))
15	13, 14	pm2.65da 817	. . . . . . . 8 ⊢ (𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) → ¬ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
16	15	nrex 3063	. . . . . . 7 ⊢ ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅
17	16	a1i 11	. . . . . 6 ⊢ (TC+ 𝐴 ∈ V → ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
18		difexg 5259	. . . . . . 7 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V)
19		zfreg 9500	. . . . . . 7 ⊢ (((TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
20	18, 19	sylan 581	. . . . . 6 ⊢ ((TC+ 𝐴 ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
21	17, 20	mtand 816	. . . . 5 ⊢ (TC+ 𝐴 ∈ V → ¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅)
22		nne 2934	. . . . 5 ⊢ (¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅ ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
23	21, 22	sylib 218	. . . 4 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
24		ssdif0 4296	. . . 4 ⊢ (TC+ 𝐴 ⊆ ∪ (𝑅1 “ On) ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
25	23, 24	sylibr 234	. . 3 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))
26		eleq1 2823	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On)))
27		sseq1 3942	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ⊆ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
28	26, 27	bibi12d 345	. . . 4 ⊢ (𝑥 = TC+ 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On)) ↔ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))))
29	28, 12	vtoclg 3497	. . 3 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
30	25, 29	mpbird 257	. 2 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
31		elex 3448	. 2 ⊢ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ V)
32	30, 31	impbii 209	1 ⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2930  ∃wrex 3059  Vcvv 3427   ∖ cdif 3882   ∩ cin 3884   ⊆ wss 3885  ∅c0 4263  ∪ cuni 4840   “ cima 5623  Oncon0 6312  𝑅₁cr1 9675  TC+ cttc 36656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-reg 9496

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-r1 9677  df-ttc 36657

This theorem is referenced by:  ttcwf3  36696